Tropical Arithmetic and Matrix Algebra

This article introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e., summation and maximum. Although our framework is combinatorial, notions of regularity and invertibility arise naturally for matrices over this semiring; we show that a tropical matrix is invertible if and only if it is regular.     

Dominance and Transmissions in Supertropical Valuation Theory

This article is a sequel of [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar]], where we defined supervaluations on a commutative semiring R and studied a dominance relation ? ≥ ψ between supervaluations ? and ψ on R, aiming at an enrichment of the algebraic tool box for use in tropical geometry.

A supervaluation ?: R → U is a multiplicative map from R to a supertropical semiring U, cf. [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar]], [7 Izhakian , Z. , Rowen , L. ( 2011 ). Supertropical matrix algebra . Israel J. Math. 182 ( 1 ): 383 – 424 .[Crossref], [Web of Science ®] , [Google Scholar]], [8 Izhakian , Z. , Rowen , L. ( 2010 ). Supertropical polynomials and resultants . J. Alg. 324 : 1860 – 1886 . (Preprint at arXiv:0902.2155.) [Crossref], [Web of Science ®] , [Google Scholar]], [5 Izhakian , Z. , Knebusch , M. , Rowen , L. Supertropical monoids: Basics and canonical factorization . Preprint at arXiv:1108.1880 . [Google Scholar]], [9 Maclane , S. ( 1998 ). Categories for the Working Mathemtician. , 4th ed. Springer Vereag . [Google Scholar]], with further properties, which mean that ? is a sort of refinement, or covering, of an m-valuation (= monoid valuation) v: R → M. In the most important case, that R is a ring, m-valuations constitute a mild generalization of valuations in the sense of Bourbaki [1 Bourbaki , N. Algèbre Commutative VI, §3 No. 1 . [Google Scholar]], while ? ≥ ψ means that ψ: R → V is a sort of coarsening of the supervaluation ?. If ?(R) generates the semiring U, then ? ≥ ψ iff there exists a “transmission” α: U → V with ψ = α ○ ?.

Transmissions are multiplicative maps with further properties, cf. [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar], Section 5]. Every semiring homomorphism α: U → V is a transmission, but there are others which lack additivity, and this causes a major difficulty. In the main body of the article we study surjective transmissions via equivalence relations on supertropical semirings. We put special emphasis on homomorphic equivalence relations. Even those are often much more complicated than congruences by ideals in usual commutative algebra.     

An upper bound on the number of Steiner triple systems

Richard Wilson conjectured in 1974 the following asymptotic formula for the number of n ‐vertex Steiner triple systems: \begin{align*} STS(n) = \left( (1+o(1))\frac{n}{e^2} \right)^{\frac{n^2}{6}}\end{align*}. Our main result is that The proof is based on the entropy method. As a prelude to this proof we consider the number F(n) of 1 ‐factorizations of the complete graph on n vertices. Using the Kahn‐Lovász theorem it can be shown that We show how to derive this bound using the entropy method. Both bounds are conjectured to be sharp. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 43, 399–406, 2013     

Categories of Layered Semirings

We generalize the constructions of layered domains^? to layered semirings, in order to enrich the structure, and in particular to provide finite examples for applications in arithmetic. The layered category theory is extended accordingly, to cover noncancellative monoids, which are examined in their own right.     

Optical transitions of Eu3+ and Dy3+ ions in lead phosphate glasses

Lead phosphate glasses containing Eu(3+) and Dy(3+) have been studied. Local structure was verified using Fourier transform (FT)-IR spectroscopy. Emission bands of Eu(3+) and Dy(3+) ions in lead phosphate glasses are observed in the visible spectral range, which correspond to 5D0→7F(J) (J=0,1,2,4) and 4F(9/2)→6H(J/2) (J=15,13,11) transitions, respectively. Shorter luminescence decays from excited states of Eu(3+) and Dy(3+0 are due to the presence of PbO in phosphate glass.     

Structural Diversity and Thermochromic Properties of Iodobismuthate Materials Containing d-Metal Coordination Cations: Observation of a High Symmetry [Bi(3)I(11)](2-) Anion and of Isolated I(-) Anions

Six new inorganic-organic salts, all containing iodobismuthate anions and d-metal coordination cations, were synthesized solvothermally from reactions of bismuth iodide, a transition metal (M) nitrate salt (M = Co, Fe or Zn), and a heterocyclic, chelating organic ligand: 1,10-phenanthroline (1,10-phen), 3,4,7,8-tetramethyl-1,10-phenanthroline (TMphen), or 2,2':6',2'-terpyridine (tpy). All six compounds were structurally analyzed by single crystal X-ray diffraction, including variable temperature crystallographic analysis to monitor for structural changes. Furthermore, those containing novel anions and achieved in high yield were additionally characterized by solid-state UV visible spectroscopy at room temperature. [Co(1,10-phen)(3)][Bi(3)I(11)] (1), [Fe(1,10-phen)(3)][Bi(3)I(11)] (2), and [Zn(1,10-phen)(3)][Bi(3)I(11)] (3) are isostructural. They crystallize in the monoclinic space group P2(1)/n and contain the unprecedented iodobismuthate anion, [Bi(3)I(11)](2-), which exhibits near D(3h) symmetry and has an unusual arrangement of three cis face-sharing BiI(6) octahedra. [Co(TMPhen)(3)](2)[Bi(2)I(9)][I] (4), which crystallizes in the trigonal space group P-31c, and [Co(tpy)(2)](2)[Bi(2)I(9)][I] (5) and [Zn(tpy)(2)](2)[Bi(2)I(9)][I] (6), which are isostructural and crystallize in the monoclinic space group C2/c, contain the discrete binuclear [Bi(2)I(9)](3-) anion, common in previously reported iodobismuthate compounds. In addition they contain unusual isolated I(-) anions, which are rarely encountered in iodobismuthate phases. Compounds 1-6 show constitutional similarities while utilizing different organic ligands and illustrate the sensitive dependence of reaction conditions on the identity of the halometalate anion formed. Additionally, all six compounds and the starting material BiI(3) are thermochromic; the origin of this behavior is spectroscopically and crystallographically investigated.     

An approximation algorithm for counting contingency tables

We present a randomized approximation algorithm for counting contingency tables, m × n non‐negative integer matrices with given row sums R = (r₁,…,r_m) and column sums C = (c₁,…,c_n). We define smooth margins (R,C) in terms of the typical table and prove that for such margins the algorithm has quasi‐polynomial N^{O(ln N)} complexity, where N = r₁ + … + r_m = c₁ + … + c_n. Various classes of margins are smooth, e.g., when m = O(n), n = O(m) and the ratios between the largest and the smallest row sums as well as between the largest and the smallest column sums are strictly smaller than the golden ratio (1 + )/2 ≈? 1.618. The algorithm builds on Monte Carlo integration and sampling algorithms for log‐concave densities, the matrix scaling algorithm, the permanent approximation algorithm, and an integral representation for the number of contingency tables. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Preparation and characterization of novel inorganic-organic hybrid materials containing rare, mixed-halide anions of bismuth(III)

Three new hybrid inorganic-organic salts containing novel mixed haloanions of bismuth were synthesized by the solvothermal reaction of bismuth iodide with a haloacid, HX (X = Cl or Br), and the alkylamine 4,4'-trimethylenedipiperidine (TMDP). All three compounds were structurally characterized by single-crystal X-ray diffraction. Reaction of TMDP and BiI(3) with HCl yielded two crystalline products: [H(2)TMDP](2)[(Bi(2)I(9))(BiCl(2)I(2))] (1, major yield) and [H(2)TMDP](2)[Bi(2)Cl(10-x)I(x)] (2, x = 3.83, minor yield). Compound 1 crystallizes in the monoclinic space group Cc (a = 22.8586(11) A, b = 15.5878(7) A, c = 17.6793(9) A, beta = 118.7010(10) degrees , Z = 4) and contains the mononuclear mixed-halide anion BiCl(2)I(2)(-) in addition to a face-sharing bioctahedral Bi(2)I(9)(3)(-) anion and two independent H(2)TMDP(2+) cations. The BiCl(2)I(2)(-) anion has a sawhorse geometry (equatorially vacant trigonal bipyramidal geometry) that is not commonly observed in bismuth chemistry. Compound 2 crystallizes in the monoclinic space group P2(1)/c (a = 14.9471(7) A, b = 12.7622(6) A, c = 13.3381(7) A, beta = 116.1030(10) degrees , Z = 2) and contains an edge-sharing bioctahedral mixed-halide anion in which iodide occupies one and chloride occupies two of the five crystallographically independent halide sites. The remaining two sites have mixed-chloride and -iodide occupancy. Reaction of TMDP and BiI(3) with HBr yielded the crystalline product [H(2)TMDP][BiBr(5-x)I(x)] (3, x = 0.99), which contains, in addition to the organic cation, a polymeric, mixed-haloanion of bismuth(III). Compound 3 crystallizes in the chiral, orthorhombic space group P2(1)2(1)2(1) (a = 8.5189(5) A, b = 14.8988(9) A, c = 17.9984(11) A, Z = 4) and consists of an H(2)TMDP(2+) cation in addition to the anion, which is built up of corner-sharing BiX(6) octahedra. Of the five crystallographically independent halide sites in this anion, two are occupied solely by Br and the remaining three have mixed-bromide and -iodide occupancy. Other anion stoichiometries have been observed crystallographically for 3, as the specific stoichiometry is dependent on the relative concentration of the haloacid starting material used.